A  is a cylinder of radius  and height  B  is a sphere with radius .

Sphere 's volume is 5 times as big as the volume of cylinder .

Find an expression for  in terms of .

We are given that the volume of the sphere \( B \) is 5 times the volume of the cylinder \( A \). We need to find an expression for the radius \( r \) of the sphere in terms of the radius \( R \) and height \( H \) of the cylinder.

### Step 1: Write the volume of the cylinder \( A \)
The formula for the volume of a cylinder is:
\[
V_{\text{cylinder}} = \pi R^2 H
\]
where \( R \) is the radius of the base of the cylinder, and \( H \) is the height of the cylinder.

### Step 2: Write the volume of the sphere \( B \)
The formula for the volume of a sphere is:
\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]
where \( r \) is the radius of the sphere.

### Step 3: Relate the volumes
We are told that the volume of the sphere is 5 times the volume of the cylinder. Therefore, we have the equation:
\[
V_{\text{sphere}} = 5 \times V_{\text{cylinder}}
\]
Substituting the formulas for the volumes:
\[
\frac{4}{3} \pi r^3 = 5 \times \pi R^2 H
\]

### Step 4: Simplify the equation
We can divide both sides of the equation by \( \pi \) to eliminate it:
\[
\frac{4}{3} r^3 = 5 R^2 H
\]

### Step 5: Solve for \( r^3 \)
Multiply both sides by \( \frac{3}{4} \) to isolate \( r^3 \):
\[
r^3 = \frac{15}{4} R^2 H
\]

### Step 6: Solve for \( r \)
Take the cube root of both sides to solve for \( r \):
\[
r = \left( \frac{15}{4} R^2 H \right)^{\frac{1}{3}}
\]

Thus, the expression for the radius \( r \) of the sphere in terms of the radius \( R \) and height \( H \) of the cylinder is:
\[
r = \left( \frac{15}{4} R^2 H \right)^{\frac{1}{3}}
\]

### Do you want more details or have any questions?

### 5 Related Questions:
1. How would the expression change if the sphere’s volume was 3 times that of the cylinder?
2. What is the volume of the sphere if \( R = 2 \) and \( H = 3 \)?
3. How does the volume of the cylinder change if both \( R \) and \( H \) are doubled?
4. Can you find the surface area of the cylinder in terms of \( R \) and \( H \)?
5. How does the relationship change if the height of the cylinder is a function of its radius, say \( H = 2R \)?

### Tip:
When dealing with volumes of 3D shapes, always make sure to pay attention to units. Using consistent units across all measurements will help avoid mistakes in calculations.

A is a cylinder of radius R and height H. B is a sphere with radius r. Sphere B's volume is 5 times as big as the volume of cylinder A. Find an expression for r in terms of R and H.

Discover how to find an expression for the radius of a sphere, where its volume is five times that of a cylinder. This problem explores volume formulas for cylinders and spheres, applying algebra to solve for the sphere's radius. Suitable for high school students in Grades 10-12.

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